Extension of Matrices with Entries in H on Coverings of Riemann Surfaces of Finite Type

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1 Extension of Matrices with Entries in H on Coverings of Riemann Surfaces of Finite Type arxiv: v1 [math.cv] 11 Jan 2008 Alexander Brudnyi Department of Mathematics and Statistics University of Calgary, Calgary Canada Abstract In the present paper continuing our work started in [Br1]-[Br5] we prove an extension theorem for matrices with entries in the algebra of bounded holomorphic functions defined on an unbranched covering of a Caratheodory hyperbolic Riemann surface of finite type. 1. Introduction. Let X be a complex manifold and let H (X) be the Banach algebra of bounded holomorphic functions on X equipped with the supremum norm. We assume that X is Caratheodory hyperbolic, that is, the functions in H (X) separate the points of X. The maximal ideal space M = M(H (X)) is the set of all nonzero linear multiplicative functionals on H (X). Since the norm of each φ M is 1, M is a subset of the closed unit ball of the dual space (H (X)). It is a compact Hausdorff space in the Gelfand topology (i.e., in the weak topology induced by (H (X)) ). Also, there is a continuous embedding i : X M taking x X to the evaluation homomorphism f f(x), f H (X). The complement to the closure of i(x) in M is called the corona. The corona problem is: given X to determine whether the corona is empty. For example, according to Carleson s celebrated Corona Theorem [C] this is true for X being the open unit disk in C. (This was conjectured by Kakutani in 1941.) Also, there are non-planar Riemann surfaces for which the corona is non-trivial (see, e.g., [JM], [G], [BD], [L] and references therein). This is due to a structure that in a sense makes the surface seem higher dimensional. So there is a hope that the restriction to the Riemann sphere might prevent this obstacle. However, the general problem for planar domains is still open, as is the Research supported in part by NSERC Mathematics Subject Classification. Primary 30D55. Secondary 30H05. Key words and phrases. Corona theorem, bounded holomorphic function, covering, Riemann surface of finite type. 1

2 problem in several variables for the ball and polydisk. (In fact, there are no known examples of domains in C n, n 2, without corona.) At present, the strongest corona theorem for planar domains is due to Moore [M]. It states that the corona is empty for any domain with boundary contained in the graph of a C 1+ǫ -function. This result is an extension of an earlier result of Jones and Garnett [GJ] for a Denjoy domain (i.e., a domain with boundary contained in R). The corona problem can be reformulated as follows, see, e.g., [Ga]: A collection f 1,...,f n of functions from H (X) satisfies the corona condition if max f j(x) δ > 0 for all x X. (1.1) 1 j n The corona problem being solvable (i.e., the corona is empty) means that the Bezout equation f 1 g f n g n 1 (1.2) has a solution g 1,...,g n H (X) for any f 1,...,f n satisfying the corona condition. We refer to max 1 j n g j as a bound on the corona solutions. (Here is the norm on H (X).) In [Br4, Theorem 1.1] using an L 2 cohomology technique we proved Theorem 1.1 Let r : X Y be a connected unbranched covering of a Caratheodory hyperbolic Riemann surface of finite type Y (i.e., the fundamental group of Y is finitely generated). Then X is Caratheodory hyperbolic and for any f 1,..., f n H (X) satisfying (1.1) there are solutions g 1,...,g n H (X) of (1.2) with the bound max 1 j n g j C(Y, n, max 1 j n f j, δ). This result extends the class of Riemann surfaces for which the corona theorem is valid (see also [Br1]). On the other hand, from the results of Lárusson [L] (sharpened in [Br3]) one obtains that the assumption of the Caratheodory hyperbolicity of Y cannot be removed. Specifically, for any integer n 2 there are a compact Riemann surface S n and its regular covering r n : S n S n such that (a) S n is a complex submanifold of an open Euclidean ball B n C n ; (b) the embedding i : S n B n induces an isometry i : H (B n ) H ( S n ). In particular, the maximal ideal spaces of H ( S n ) and H (B n ) coincide. The main result of our paper is the following noncommutative analog of the above theorem: Theorem 1.2 Let r : X Y satisfy the assumptions of Theorem 1.1 and A = (a ij ) be an n k matrix, k < n, with entries in H (X). Assume that the family of minors of order k of A satisfies the corona condition. Then there is an n n matrix à = (ã ij ), ã ij H (X), so that ã ij = a ij for 1 j k, 1 i n, and det à = 1. Moreover, the corresponding norm of à is bounded by a constant depending on the norm of A, δ (from (1.1) for the family of minors of order k of A), n and Y only. 2

3 Previously, a similar result was proved for matrices with entries in H (U) for domains U X such that the embedding induces an injective homomorphism of the corresponding fundamental groups and r(u) Y, see [Br2, Theorem 1.1]. Its proof was based on a Forelli type theorem on projections in H (see [Br1]) and a Grauert type theorem for holomorphic vector bundles on maximal ideal spaces (which are not usual manifolds) of certain Banach algebras (see [Br2]) along with some ideas of Tolokonnikov [T] (see also this paper for further results and references on the extension problem for matrices with entries in different function algebras). The remarkable class of Riemann surfaces X for which a Forelli type theorem and, hence, the corona theorem are valid was introduced by Jones and Marshall [JM]. The definition is in terms of an interpolating property for the critical points of the Green function on X. It is an interesting open question whether the result analogous to Theorem 1.2 is valid for such X. 2. Auxiliary Results For a set of indices Λ consider the family X Λ := {X λ } λ Λ where each X λ is a connected unbranched covering of Y. By r λ := X λ Y we denote the corresponding projection. Considering this family as the disjoint union of sets X λ we introduce the natural complex structure on X Λ. Thus r Λ : X Λ Y is an unbranched covering of Y where r Λ Xλ := r λ. We say that a function f on X Λ belongs to H (X Λ ) if f Xλ H (X λ ), λ Λ, and sup λ Λ f Xλ <. Proposition 2.1 The corona theorem is valid for H (X Λ ). Proof. Let f 1,...,f n H (X Λ ) satisfy the corona condition (1.1). We set f jλ := f j Xλ. Then each family f 1λ,...,f nλ H (X λ ) satisfies (1.1) with the same δ as for f 1,...,f n. According to Theorem 1.1 there are functions g 1λ,...,g nλ H (X λ ) such that f 1λ g 1λ + + f nλ g nλ 1 and max g jλ C(Y, n, max f j H 1 j n 1 j n (X Λ ), δ). Let us define g 1,...,g n H (X Λ ) by the formulas g j Xλ := g jλ. Then g 1 f g n f n 1. Let M Λ be the maximal ideal space of the Banach algebra H (X Λ ). According to Theorem 1.1, H (X Λ ) separates the points of X Λ. Thus X Λ can be regarded as a subset of M Λ. Now, by Proposition 2.1, X Λ is dense in M Λ in the Gelfand topology. We will show that Theorem 1.2 follows directly from 3

4 Theorem 2.2 Let A = (a ij ) be an n k matrix, k < n, with entries in H (X Λ ). Assume that the family of minors of order k of A satisfies the corona condition. Then there is an n n matrix à = (ã ij), ã ij H (X Λ ), so that ã ij = a ij for 1 j k, 1 i n, and det à = We recall some constructions and results presented in [Br2]. According to a construction of [Br2, section 2] the covering r Λ : X Λ Y can be considered as a fibre bundle over Y with a discrete fibre F Λ, where F Λ is the disjoint union of the fibres F λ of the coverings r λ : X λ Y, λ Λ. Let l (F Λ ) be the Banach algebra of bounded complex-valued functions f on the discrete set F Λ with pointwise multiplication and norm f = sup x FΛ f(x). Let βf Λ be the Stone-Čech compactification of F Λ, i.e., the maximal ideal space of l (F Λ ) equipped with the Gelfand topology. Then F Λ is naturally embedded into βf Λ as an open dense subset, and the topology on F Λ induced by this embedding coincides with the original one, i.e., is discrete. Every function f l (F Λ ) has a unique extension ˆf C(βF Λ ). Further, any homeomorphism φ : F Λ F Λ determines an isometric isomorphism of Banach algebras φ : l (F Λ ) l (F Λ ). Therefore φ can be extended to a homeomorphism ˆφ : βf Λ βf Λ. From here, taking closures in βf λ of fibres of the bundle r Λ : X Λ Y, we obtain a fibre bundle ˆr Λ : E(Y, βf Λ ) Y with fibre βf Λ so that X Λ is an open dense subset of E(Y, βf Λ ) (in fact, an open subbundle of E(Y, βf Λ )) and ˆr Λ XΛ = r Λ. Moreover, it was proved in [Br2, Proposition 2.1] that (1) for every h H (X Λ ) there is a unique ĥ C(E(Y, βf Λ)) such that ĥ X Λ = h. Also, it was proved in [Br4, Theorem 1.5] that for every x Y and every λ Λ the sequence r 1 λ (x) X λ is interpolating for H (X λ ) with the constant of interpolation bounded by a number depending on x and Y only. This immediately implies that (2) for each f l (rλ 1 (x)) there is a function f H (X Λ ) such that f r 1 (x) = f. In particular, the continuous extension of the algebra H (X Λ ) to E(Y, βf Λ ) separates the points on E(Y, βf Λ ). Thus E(Y, βf Λ ) can be regarded as a dense subset of M Λ. Let (U i ) i I be a countable cover of Y by compact subsets U i Y homeomorphic to a closed ball in R 2. Then by our construction Ûi := ˆr Λ 1 (U i ) is homeomorphic to U i βf Λ. So, E(Y, βf Λ ) is a countable union of compact subsets Ûi. Since the covering dimension dim Ûi of Ûi is 2, i I, this implies (cf. [Br2, Proposition 4.1]) Λ (3) dime(y, βf Λ ) = 2. Taking now an open countable cover of Y by relatively compact subsets homeomorphic to an open ball in R 2 and the corresponding open cover of E(Y, βf Λ ) by their preimages with respect to ˆr Λ we get (4) E(Y, βf Λ ) is an open dense subset of M Λ, and the restriction of the Gelfand topology on M Λ to E(Y, βf Λ ) coincides with the topology of E(Y, βf Λ ). 4

5 2.3. Since Y is a Riemann surface of finite type, the theorem of Stout [St, Theorem 8.1] implies that there exist a compact Riemann surface R and a holomorphic embedding φ : Y R such that R \ φ(y ) consists of finitely many closed disks with analytic boundaries together with finitely many isolated points. Since Y is Caratheodory hyperbolic, the set of the disks in R \ φ(y ) is not empty. Also, without loss of generality we may and will assume that the set of isolated points in R \ φ(y ) is not empty, as well. (For otherwise, φ(y ) is a bordered Riemann surface and the required result follow from [Br2, Theorem 1.1].) We will naturally identify Y with φ(y ). Also, we set R \ Y := D i 1 i k 1 j l {x j } and Z := Y 1 j l {x j } (2.1) where each D i is biholomorphic to the open unit disk D C and these biholomorphisms are extended to diffeomorphisms of the closures D i D. Then Z R is a bordered Riemann surface with a nonempty boundary. In particular, there is a bordered Riemann surface Z containing Z such that Z is a deformation retract of Z. We set Y := Z \ {x 1,...,x l }. (2.2) Then Y Y and π 1 (Y ) = π 1 (Y ) (here π 1 (M) stands for the fundamental group of M). This implies that for each λ Λ there is a connected covering X λ of Y such that X λ is an open connected subset of X λ. Without loss of generality we denote the covering projection X λ Y by the same symbol r λ (as for X λ ). Now, we define X Λ := {X λ } λ Λ so that X Λ is an open subset of X Λ and r Λ : X Λ Y, r Λ X λ := r λ. Further, similarly to the constructions of section 2.2 we determine the bundle ˆr Λ : E(Y, βf Λ ) Y so that E(Y, βf Λ ) is an open subbundle of E(Y, βf Λ ). Then X Λ and E(Y, βf Λ ) possess the properties similar to (1)-(3) for X Λ and E(Y, βf Λ ). Let cl(y ) denote the closure of Y in Y. We set Then we have E(cl(Y ), βf Λ ) := ˆr 1 Λ (cl(y )). (5) dim E(cl(Y ), βf Λ ) = 2 and E(Y, βf Λ ) E(cl(Y ), βf Λ ) is an open dense subset By H (E(Y, βf Λ )) we denote the extension of H (X Λ ) to E(Y, βf Λ ) described in section 2.2. We will use also the algebra H (E(Y, βf Λ )) determined similarly (i.e., with Y and X Λ substituted for Y and X Λ). Next, let us consider Banach subalgebras A 1, A 2 of H (E(Y, βf Λ )) defined as follows. A 1 := {ˆr Λ f H (E(Y, βf Λ )) : f H (Z )}. (2.3) (Here ˆr Λ f is the pullback of f with respect to ˆr Λ.) To define A 2 we choose a function φ H (Z ) with the set of zeros {x 1,..., x l } so that each x j is a zero of order 1 of φ. (Since Z R is a bordered Riemann 5

6 surface with a nonempty boundary, such a φ exists due to [Br2, Corollary 1.8].) Then A 2 is the uniform closure of the algebra of functions f H (E(Y, βf Λ )) of the form f := g + ([ˆr Λ φ] h) E(Y,βF Λ ), g A 1, h H (E(Y, βf Λ )). (2.4) By the definition A 2 separates the points of E(Y, βf Λ ) (because H (E(Y, βf Λ )) separates the points of E(Y, βf Λ ) and ˆr Λφ is nonzero on the fibres of ˆr Λ ). Clearly, we have embeddings i A 1 i 1 2 A2 H (E(Y, βf Λ )). (2.5) The transpose maps to these embeddings determine continuous surjective maps i M 2 i Λ M 1 2 M 1 (2.6) where M 2 is the closure in the Gelfand topology of the image of E(Y, βf Λ ) in the maximal ideal space of A 2, and M 1 is the closure in the Gelfand topology of the image of E(Y, βf Λ ) in the maximal ideal space of A 1. (Here we used that the closure in the Gelfand topology of E(Y, βf Λ ) M Λ is M Λ, see Proposition 2.1.) By the definition, M 1 = Z and E(cl(Y ), βf Λ ) M 2 (see section 2.3). Moreover, the restriction of i 2 to E(Y, βf Λ ) is the identity map and the restriction of i 1 to E(cl(Y ), βf Λ) can be naturally identified with ˆr Λ so that (i 1 ) 1 (cl(y )) = ˆr Λ 1 (cl(y )) = E(cl(Y ), βf Λ). Now, we prove Lemma 2.3 For each x j M 1 the compact set (i 1) 1 (x j ) consists of a single point (which we naturally identify with x j ), 1 j l. Proof. Let {ξ 1,α }, {ξ 2,α } E(Y, βf Λ ) be nets converging to points ξ 1, ξ 2 (i 1 ) 1 (x j ). Then for f from (2.4) and i = 1, 2 we have f(ξ i ) = lim α f(ξ i,α ) = lim α (g(ξ i,α ) + (ˆr Λφ)(ξ i,α ) h(ξ i,α )) := g(x j ). (2.7) (We used here that the nets {i 1 (ξ 1,α )}, {i 1 (ξ 2,α )} M 1 converge to x j.) This implies that ξ 1 = ξ 2. Corollary 2.4 dim M 2 = 2. Proof. According to Lemma 2.3 and property (5) of section 2.3, M 2 is the disjoint union of zero-dimensional sets {x j }, 1 j l, and the two-dimensional set E(cl(Y ), βf Λ ). Hence dimm 2 = 2, see, e.g., [N, Chapter 2, Theorem 9-11] We fix coordinate neighbourhoods N j Z (see (2.1)) biholomorphic to D of points x j, 1 j l, and a bordered Riemann surface S Y such that N i N j = for i j, each Y N j does not contain x j and is biholomorphic to an 6

7 annulus and U := S ( 1 j l N j ) is an open cover of Y. Here N j := N j \ {x j } is biholomorphic to D := D \ {0}. We set NjΛ := r 1 Λ (N j ), 1 j l, and S Λ := rλ 1 (S). (2.8) Let V X Λ be either one of N jλ or S Λ. By H (V ) we denote the Banach algebra of bounded holomorphic functions on V defined as in section 2.1 for X Λ. Further, we set ˆN j := (i 2 i 1) 1 (N j ), 1 j l, Ŝ := (i 2 i 1) 1 (S Z) (2.9) Here Z is the boundary of the bordered Riemann surface Z that can be regarded as the outer boundary of S. By the definition ˆN j, 1 j l, and Ŝ are open subsets of M Λ forming a cover of this space. The main fact used in the proof of Theorem 1.1 is Proposition 2.5 Assume that f H (V ) where V is either one of N jλ or S Λ. Then f admits a continuous extension ˆf to ˆV where ˆV stands for the corresponding ˆN j or Ŝ. Proof. First, we will prove the result for V = N jλ. Let ρ j be a C -function on R equal to 1 in a neighbourhood of x j with supp(ρ) N j. We set f 1 := (r Λρ j ) f. (2.10) Then f 1 can be considered as a C -function on X Λ (defined in section 2.3). Further, we introduce a (0, 1)-form on X Λ by the formula ω := f 1 ρ Λφ. (2.11) The definition is correct because f 1 equals 0 on ρ 1 Λ (O) for some neighbourhood O of x j and on X Λ \ N jλ, and ρ Λ φ 0 on N jλ. Thus ω is a -closed 1-form on X Λ. Consider the form ω λ := ω X λ on X λ. Let us assume that Z is equipped with a hermitian metric h Z with the associated (1, 1)-form ω Z. Then we equip X λ with the hermitian metric h X λ induced by the pullback rλ ω Z of ω Z to X λ. Now, if η is a smooth (0, 1)-form on X λ, by η z, z X λ, we denote the norm of η at z defined by the hermitian metric h X on the fibres of λ the cotangent bundle T X λ on X λ. Next, since f H (NjΛ) and r Λ (supp(ω)) =: K Y, see (2.2), one easily obtains from (2.11) that ω := sup λ Λ { sup z X λ ω λ z } < (2.12) 7

8 From here by [Br4, Theorem 1.6] we obtain that the equation g λ = ω λ has a smooth bounded solution g λ on X λ such that g λ L := sup g λ (z) C ω (2.13) z X λ with C depending on K, Z and h Z only. We define bounded functions g and f 2 on X Λ by the formulas g X λ := g λ, λ Λ, f 2 := (ρ Λφ) g. (2.14) Then we have (a) f 2 = f 1 on X Λ and (b) lim α f 2 (ξ α ) = 0 (2.15) for each net {ξ α } X Λ such that {r Λ(ξ α )} Y is a net converging to any x s, 1 s l. In particular, f 3 := f 1 f 2 H (X Λ ). (2.16) Thus f 3 admits a continuous extension ˆf 3 to M Λ. Let us prove now that (*) f 2 admits a continuous extension ˆf 2 to M Λ. Indeed, by the definition of f and r Λ ρ j, the function f 1 defined by (2.10) has a continuous extension to E(Y, βf Λ ), see [Br2, Proposition 2.1]. Thus, f 2 := f 1 f 3 admits a continuous extension to E(Y, βf Λ ), as well. (We denote this extension also by f 2.) Now if {ξ α } E(Y, βf Λ ) is a net converging to a point ξ M 2 (see (2.6)) such that i 1(ξ) = x s for some 1 s l, then from (2.15) (b) we get lim α f 2 (ξ α ) = 0. Since (i 1 ) 1 (x s ) = x s, the latter implies that the function f 2 equals 0 at each x s and f 2 on E(cl(Y ), βf Λ ) is continuous on M 2. Therefore the function ˆf 2 := i f 2 2 is continuous on M Λ. Since the restriction of i 2 to E(Y, βf Λ ) is the identity map, ˆf2 is a continuous extension of f 2. This proves (*). From (2.16) and (*) we obtain that f 1 admits a continuous extension ˆf 1 to M Λ. Now, ˆNj from (2.9) is the union of ˆr Λ 1 (Nj ) E(Y, βf Λ) and (i 2 i 1 ) 1 (O) where O N j is a neighbourhood of x j such that ρ j 1 on O. The function f admits a continuous extension f on ˆr Λ 1 (Nj ), see [Br2, Proposition 2.1], and f = f 1 on rλ 1 (O). Thus the function ˆf defined by ˆf := ˆf 1 on (i 2 i 1 ) 1 (O) and ˆf := f on ˆr 1 Λ (N j ) is the required continuous extension of f to ˆN j. Finally, in the case V = S Λ we choose a C -function ρ on R equals 0 on Y \ S and 1 on R \ Z with supp(dρ) S. Then repeating the above arguments with ρ j substituted for ρ we obtain the proof of the proposition in this case. We leave the details to the readers. 8

9 3. Proof of Theorem Proof of Theorem 2.2. Let A = (a ij ) be an n k matrix, k < n, with entries in H (X Λ ). Assume that the family of minors of order k of A satisfies the corona condition (1.1). Due to the corona theorem for H (X Λ ), see Proposition 2.1, we can extend A continuously to M Λ such that the family of minors of order k of the extended matrix  = (â ij) satisfies (1.1) on M Λ with the same δ as for A. Next, according to [L, Theorem 3], to prove the theorem it suffices to find an n n matrix B = (b ij ), b ij C(M Λ ), so that b ij = â ij for 1 j k, 1 i n, and detb = 1. Note that the matrix  determines a trivial subbundle ξ of complex rank k in the trivial vector bundle θ n := M Λ C n on M Λ. Let η be an additional to ξ subbundle of θ n, i.e., ξ η = θ n. We will prove that η is topologically trivial. Then a trivialization s 1, s 2,...,s n k C(M Λ, η) (given by global continuous sections of η) will determine the required continuous extension B of the matrix Â. Let us prove first that  can be extended to an invertible matrix on each ˆN j and Ŝ, see (2.9). Lemma 3.1 Let ˆV be either one of ˆN j or Ŝ. Then for  ˆV there is an n n matrix BˆV = (b ij;ˆv ), b ij;ˆv C(ˆV ), so that b ij;ˆv = â ij ˆV for 1 j k, 1 i n, and det BˆV = 1. Moreover, BˆV V has entries in H (V ) where V := ˆV X Λ. Proof. First assume that ˆV = ˆN j so that V = NjΛ is an unbranched covering of Nj = D, see (2.8). Then by the definition NjΛ = {N jλ } λ Λ where each Njλ := NjΛ X λ is an unbranched covering of Nj consisting of at most countably many connected components. Thus each Njλ is biholomorphic to k K λ W jλ;k, K λ N, where each W jλ;k is either D or D. Now, A Wjλ;k satisfies conditions of Theorem 1.1 with the same δ as for A. According to the main result of Tolokonnikov [T] for H -matrices on D, there is a matrix B jλ;k with entries in H (W jλ;k ) which extends A Wjλ;k in the sense of Theorem 1.1 and such that detb jλ;k = 1 and sup B jλ;k C (3.1) j,λ,k where C depends on the norm of A on X Λ, δ and n. (Here for a matrix C = (c ij ) with entries in H (O) we set C := max i,j c ij H (O).) In particular, (3.1) implies that the matrix B jλ on NjΛ defined by B jλ Wjλ;k = B jλ;k, 1 j l, k K λ, λ Λ, has entries in H (N jλ ), extends A N jλ and det B jλ = 1. According to Proposition 2.5, B jλ is extended to a continuous matrix B ˆNj on ˆN j. This matrix extends  ˆNj and satisfies the required conditions of the lemma. Consider now the case ˆV = Ŝ so that V = S Λ is an unbranched covering of S, see (2.8). In this case we apply similar to the above arguments where instead of the result of [T] we use [Br2, Theorem 1.1] applied to the coverings S λ := S Λ X λ of a bordered Riemann surface S. Then we obtain a matrix B Λ on S Λ with entries in 9

10 H (S Λ ) which extends A SΛ and such that detb Λ = 1. Applying again Proposition 2.5, we extend B Λ continuously to Ŝ so that the extended matrix B Ŝ satisfies the required conditions of the lemma. Let ξ q be the quotient bundle of θ n with respect to the subbundle ξ. By the definition ξ q is isomorphic (in the category of continuous bundles on M Λ ) to η. Thus it suffices to prove that ξ q is topologically trivial. Now, Lemma 3.1 implies straightforwardly that ξ q ˆV is topologically trivial for ˆV being either one of ˆN j or Ŝ. In particular, ξ q is defined by a 1-cocycle defined on the open cover { ˆN 1,..., ˆN l, Ŝ} of M Λ (see, e.g., [H] for the general theory of vector bundles). Since by the definition ˆN i ˆN j = for i j, this cocycle consists of continuous matrix-functions C i C( ˆN i Ŝ, GL n k(c)), 1 i l. Set now Ñj := (i 1 ) 1 (N j ), S := (i 1 ) 1 (S), see (2.6). Then {Ñ1,...,Ñl, S} is an open cover of M 2. Moreover, the map i 2 : M Λ M 2 is identity on each ˆN i Ŝ, see section 2.4. Therefore each C i can be regarded as a matrix-function on Ñi S. In particular, these functions determine a complex vector bundle ξ q of rank n k on M 2 so that i 2 ξ q = ξ q. (3.2) Since dim M 2 = 2, see Corollary 2.4, the bundle ξ q is isomorphic to θm n k 1 2 θ := M 2 C n k 1 is the trivial bundle and θ is a vector bundle of where θ n k 1 M 2 complex rank 1, see, e.g., [Br5, Lemma 2.8]. This and (3.2) imply that ξ q = η is isomorphic to θ n k 1 i 2θ where θ n k 1 = M Λ C n k 1 is the trivial bundle. Now, for the first Chern classes (which are additive with respect to the operation of the direct sum of bundles) we have the following identity 0 = c 1 (θ n ) = c 1 (ξ) + c 1 (η) = c 1 (θ n k 1 i 2 θ) = c 1(i 2θ). (3.3) We used here that Chern classes of trivial bundles are zeros. Equality (3.3) shows that the first Chern class of the complex rank 1 vector bundle i 2θ is zero. Thus this bundle is topologically trivial (see, e.g., [H]). Combining this fact with the above isomorphism for η we get η = θ n k := M Λ C n k. This completes the proof of Theorem Proof of Theorem 1.1. Let us define Λ Y ;M,δ as the set of all possible couples (A, X) where X is a connected covering of Y and A is an n k matrix on X satisfying conditions of Theorem 1.1 with a fixed δ in the corona condition (1.1) for the family of minors of order k and such that A M. For Λ := Λ Y ;M,δ we consider the n k matrix A with entries in H (X Λ ) defined as follows A Xλ := A, λ = (A, X) Λ, X λ := X. (3.4) Then clearly A satisfies conditions of Theorem 2.2 on X Λ. According to this theorem there is an n n matrix à with entries in H (X Λ ) and with detã = 1 that extends A. For λ = (A, X) Λ we set à := à X. 10

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